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Optimal Encodings for Range Majority Queries

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Abstract

We study the problem of designing a data structure that reports the positions of the distinct \(\tau \)-majorities within any range of an array \(A[1,n]\), without storing \(A\). A \(\tau \)-majority in a range \(A[i,j]\), for \(0<\tau < 1\), is an element that occurs more than \(\tau (j-i+1)\) times in \(A[i,j]\). We show that \(\Omega (n\lceil \log (1/\tau )\rceil )\) bits are necessary for any data structure just able to count the number of distinct \(\tau \)-majorities in any range. Then, we design a structure using \(O(n\lceil \log (1/\tau )\rceil )\) bits that returns one position of each \(\tau \)-majority of \(A[i,j]\) in \(O((1/\tau )\log \log _w(1/\tau )\log n)\) time, on a RAM machine with word size \(w\) (it can output any further position where each \(\tau \)-majority occurs in \(O(1)\) additional time). Finally, we show how to remove a \(\log n\) factor from the time by adding \(O(n\log \log n)\) bits of space to the structure.

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Notes

  1. 1.

    Or an equivalent array where each element is replaced by an identifier in \([1,n]\).

  2. 2.

    Bounding \(\lg (k!)\) with integrals one obtains \(k \lg (k/e) + 1 \le \lg (k!) \le (k+1)\lg ((k+1)/e)+1\).

  3. 3.

    We could also afford to store them in plain form, in \(O((1/\tau )(\lceil \log (1/\tau )\rceil +\log \log n))\) bits.

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Acknowledgments

We thank the reviewers for their valuable comments.

Author information

Correspondence to Gonzalo Navarro.

Additional information

An early version of this article appeared in Proc. CPM 2014 [19].

Gonzalo Navarro: Partially funded by Millennium Nucleus Information and Coordination in Networks ICM/FIC P10-024F, Chile.

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Cite this article

Navarro, G., Thankachan, S.V. Optimal Encodings for Range Majority Queries. Algorithmica 74, 1082–1098 (2016). https://doi.org/10.1007/s00453-015-9987-8

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Keywords

  • Range majority queries
  • Encoding data structures
  • Succinct data structures